A Note on Stability of a Linear Functional Equation of Second Order Connected with the Fibonacci Numbers and Lucas Sequences
© Janusz Brzdęk and Soon-Mo Jung. 2010
Received: 26 April 2010
Accepted: 15 July 2010
Published: 2 August 2010
We prove the Hyers-Ulam stability of a second-order linear functional equation in single variable (with constant coefficients) that is connected with the Fibonacci numbers and Lucas sequences. In this way we complement, extend, and/or improve some recently published results on stability of that equation.
in the case where and for . The result on stability (see [5, Theorem ]) can be stated as follows.
If and , then solutions of the difference equation (1.4) are called the Lucas sequences (see, e.g., ); in some special cases they are called with specific names; for example; the Fibonacci numbers ( , , and ), the Lucas numbers ( , , and ), the Pell numbers ( , , and ), the Pell-Lucas (or companion Lucas) numbers ( , , and ), and the Jacobsthal numbers ( , , and ).
For some information and further references concerning the functional equations in single variable we refer to [15–17]; for an ample survey on stability results for those equations see . Let us mention yet that the problem of stability of functional equations is connected to the notions of controlled chaos (see ) and shadowing (see [19–21]).
Clearly, (1.1) is the characteristic equation of (1.6).
Theorem 1.3 appears to be much more general than Theorem 1.1 (obtained by a different method of proof). But on the other hand, estimation (1.5) is significantly sharper than (1.8) in numerous cases (take, e.g., and , with some large ). Therefore, there arises a natural question if the method applied in  can be modified so as to prove a more general equivalent of Theorem 1.3, but with an estimation better than (1.8). In this paper, we show that this is the case. Namely, we prove the following.
Note that, for bijective , Theorem 1.4 improves estimation (1.8) in some cases (take, e.g., , , or , ); however, in some other situations (e.g., , ), estimation (1.8) is better. Theorem 1.4 also complements Theorem 1.3 because can be quite arbitrary in the case of .
2. Proof of Theorem 1.4
In the case where is bijective, the uniqueness of results from [22, Proposition ], in view of Remark 1.2.
Hence, by [22, Proposition ], , which yields .
3. Consequences of Theorem 1.4
with . Hence, according to [22, Theorem ], there exists a function satisfying (1.6) and inequality (3.1), with . Now, it is enough to note that is a solution to (1.2), as well.
Finally, if (iii) holds, then it is enough to use [22, Theorem ] and Theorem 1.4 (the case of ).
The statement concerning uniqueness results from [22, Proposition ].
If for some (or, equivalently, for some ), then (1.2) can be nonstable, by which we mean that there is a function such that (1.7) holds with some real and for each solution of (1.2) (see, e.g., , [22, Example ], or ).
4. Some Critique and Final Remarks
which corresponds to (1.6). The second obstacle is connected with the restrictions on . For some interesting cases (Fibonacci, Lucas, or Pell numbers), we have (or, if somebody prefers, ) and such case is not covered if is not bijective (which is the case when and or, equivalently, ). All these obstacles can be overcome if, instead of Theorems 1.1–3.1, we use the following result derived from [29, Theorem ].
If and (the case of Jacobsthal numbers), then one of the roots of (1.1) is equal to and therefore (4.1) is not stable (see ), by which we mean that, for each , there is such that and for every solution of (1.6); for such function can be chosen with, for example, and (in [28, the proofs of Lemma and Theorem ] take ).
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education, Science, and Technology (no. 2010-0007143).
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